Chemistry Reference
In-Depth Information
Z
t
obs
A
¼ð
1
=
t
obs
Þ
A
ðf~
r
i
ð
t
ÞgÞ
d
t
;
t
obs
!1;
(33)
0
of observables
A
ðf
r
k
ð
t
ÞgÞ
in the system are equivalent to ensemble averages in the
microcanonical
ðN
VE
Þ
ensemble [
17
], where
E
is the total internal energy of the
system:
ðf
~
A
¼h
A
r
i
gÞi
NVE
:
(34)
The fact that the microcanonical ensemble average appears here is, of course,
due to the fact that the total energy
E
is conserved for (
32
). In practice, however, the
numerical integration of (
32
) is not exact and one has to discretize the time axis in
terms of finite time steps
t
. Thus, errors may accumulate that violate the conser-
vation law for energy in an undesirable way. These cumulative errors cannot be
suppressed entirely, but minimized using symplectic integration schemes [
217
],
such as the Verlet algorithm [
81 83
,
87
], in which the system coordinates
D
f~
r
i
ð
t
Þg
are propagated as follows:
2
4
~
~
Þ
~
Þþ
a
i
ð
r
i
ð
t
þ D
t
Þ¼
2
r
i
ð
t
r
i
ð
t
D
t
t
ÞðD
t
Þ
þOððD
t
Þ
Þ ;
(35)
where a
i
ð
ðf
~
t
Þ¼r
U
r
i
ð
t
ÞgÞ=
m
i
denotes the acceleration that acts at the
i
th
particle at time
t
. Of course,
t
in (
35
) has to be kept small enough to reach
sufficient accuracy (for an atomistic model, “small enough” means a
D
D
t
in the
range of 1 2 fs, i.e., 10
15
s!).
A useful modification of (
35
) is the so-called Velocity Verlet algorithm. It
explicitly incorporates the velocity v
i
ð
t
Þ
of the particle:
1
2
a
i
ð
2
r
i
ð
t
þ D
t
Þ¼
r
i
ð
t
Þþ
v
i
ð
t
ÞD
t
þ
t
ÞðD
t
Þ
;
(36)
1
2
a
i
ð
~
Þ¼
~
Þþ
a
i
ð
v
i
ð
t
þ D
t
v
i
ð
t
Þþ
½
t
t
þ D
t
Þ
D
t
; :
(37)
This algorithm produces integration errors of the same order as the original
Verlet algorithm. Its advantage lies in symmetric coordinates for “past” and “future”,
and it also conserves the phase space volume; i.e., Liouville´s theorem [
17
]isobeyed.
Although energy is not conserved perfectly on a short time scale, there are no
systematic energy drifts for large time scales. There exist further suggestions (e.g.,
the “leapfrog method” [
218
]) or other algorithms such as predictor corrector meth-
ods [
219
] that are more accurate at short times but that violate Liouville´s theorem.
As these methods are not symplectic, they are less in use today. We also note that
rigid constraints (rigid bond lengths, rigid bond angles, etc.) also require different
algorithms [
87
,
218
], but this topic is not considered here.
We rather focus on another aspect, namely the desirable choice of statistical
ensemble. Although statistical mechanics [
17
] asserts that in the thermodynamic
